Field With One Element - History

History

In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes. One of the assumptions is a non-triviality condition: If the building is an n-dimensional abstract simplicial complex, and if k < n, then every k-simplex of the building must be contained in at least three n-simplices. This is analogous to the condition in classical projective geometry that a line must contain at least three points. However, there are degenerate geometries which satisfy all the conditions to be a projective geometry except that the lines admit only two points. The analogous objects in the theory of buildings are called apartments. Apartments play such a constituent role in the theory of buildings that Tits conjectured the existence of a theory of projective geometry in which the degenerate geometries would have equal standing with the classical ones. This geometry would take place, he said, over a field of characteristic one. Using this analogy it was possible to describe some of the elementary properties of F₁, but it was not possible to construct it.

A separate inspiration for F₁ came from algebraic number theory. Weil's proof of the Riemann hypothesis for curves over finite fields started with a curve C over a finite field k, took its product C ×_k C, and then examined its diagonal. If the integers were a curve over a field, the same proof would prove the Riemann hypothesis. The integers Z are one dimensional, which suggests that they may be a curve, but they are not an algebra over any field. One of the conjectured properties of F₁ is that Z should be an F₁-algebra. This would make it possible to construct the product Z ×_F1 Z, and it is hoped that the Riemann hypothesis for Z can be proved in the same way as the Riemann hypothesis for a curve over a finite field.

Another angle comes from Arakelov geometry, where Diophantine equations are studied using tools from complex geometry. The theory involves complicated comparisons between finite fields and the complex numbers. Here the existence of F₁ is useful for technical reasons.

By 1991, Alexander Smirnov had taken some steps towards algebraic geometry over F₁. He introduced extensions of F₁ and used them to handle P1 over F₁. Algebraic numbers were treated as maps to this P1, and conjectural approximations to the Riemann–Hurwitz formula for these maps were suggested. These approximations imply very profound assertions like the abc conjecture. The extensions of F₁ later on were denoted as F_q with q = 1n.

In 1993, Yuri Manin gave a series of lectures on zeta functions where he proposed developing a theory of algebraic geometry over F₁. He suggested that zeta functions of varieties over F₁ would have very simple descriptions, and he proposed a relation between the K-theory of F₁ and the homotopy groups of spheres. This inspired several people to attempt to construct F₁. In 2000, Zhu proposed that F₁ was the same as F₂ except that the sum of one and one was one, not zero. Deitmar suggested that F₁ should be found by forgetting the additive structure of a ring and focusing on the multiplication. Toën and Vaquié built on Hakim's theory of relative schemes and defined F₁ using symmetric monoidal categories. Nikolai Durov constructed F₁ as a commutative algebraic monad. Soulé constructed it using algebras over the complex numbers and functors from categories of certain rings. Borger used descent to construct it from the finite fields and the integers.

Recently, Alain Connes, Caterina Consani and Matilde Marcolli have connected F₁ with noncommutative geometry.